Assume that $m, n \in \mathbb{N}$, and $T: M_{m \times n}(\mathbb{R}) ightarrow M_{m \times n}(\mathbb{R})$ is defined as $T(A)=B$ where $$ B_{i j}=\left\{\begin{array}{1} A_{i j}+A_{i(j-1)}, \text { if } j-i=1 A_{i j}, \text { otherwise } \end{array} ight. $$ Is $T$ diagonalizable? Justify your answer. CS.VS. 1425